Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice.
Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 2, Problem 101a
Solve each equation. 5 - 12x = 8 - 7x - [6 ÷ 3(2 + 53) + 5x]
Verified step by step guidance1
Simplify the expression inside the brackets first. Start with the exponentiation: \(5^3 = 125\).
Next, simplify the parentheses: \(2 + 125 = 127\). Then, divide: \(6 \div 3 = 2\). Multiply: \(2 \times 127 = 254\). The expression inside the brackets becomes \(254 + 5x\).
Substitute the simplified expression back into the equation: \(5 - 12x = 8 - 7x - (254 + 5x)\). Distribute the negative sign across the terms inside the brackets: \(5 - 12x = 8 - 7x - 254 - 5x\).
Combine like terms on the right-hand side: \(8 - 254 = -246\) and \(-7x - 5x = -12x\). The equation becomes \(5 - 12x = -246 - 12x\).
Eliminate \(-12x\) from both sides by adding \(12x\) to both sides: \(5 = -246\). This simplifies to a contradiction, indicating no solution exists for this equation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Solving Linear Equations
Solving linear equations involves isolating the variable on one side of the equation to find its value. This typically requires combining like terms, applying inverse operations, and maintaining the equality of both sides. Understanding how to manipulate equations is crucial for finding the solution.
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Order of Operations
The order of operations is a set of rules that dictates the sequence in which mathematical operations should be performed to ensure consistent results. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) helps remember this order. Properly applying these rules is essential when simplifying expressions within equations.
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Distributive Property
The distributive property states that a(b + c) = ab + ac, allowing for the multiplication of a single term across terms within parentheses. This property is vital for simplifying expressions and solving equations, especially when dealing with terms that involve variables and constants. Mastery of this concept aids in breaking down complex equations into manageable parts.
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Related Practice
Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice.
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Textbook Question
Solve each equation in Exercises 96–102 by the method of your choice. -4|x+1| + 12 = 0
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Textbook Question
Use interval notation to represent all values of x satisfying the given conditions. y = 8 - |5x + 3| and y is at least 6
Textbook Question
In Exercises 101–106, solve each equation.
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Textbook Question
Exercises 100–102 will help you prepare for the material covered in the next section. Factor: